Any useful study of a complex system requires the use of models which show how the system evolves over time. As a result, after Poincaré, little further progress was made in the study of such systems until the advent of the computer. Around that time, the term Chaos Theory was introduced to give a name to the study of complex systems ranging from natural systems such as the weather to purely mathematical concepts such as Julia Sets and the Mandelbrot Set.
For a time, Chaos Theory seemed to thrive in the media. Pictures of the Mandelbrot Set were used to popularise the subject. However, media interest soon faded for two main reasons:
- While Newtonian science led us to believe in a ‘clockwork universe’ where everything is predictable and where scientific study gives us the ability to develop many useful technologies, Chaos Theory’s initial innovation was the rather undesirable suggestion that there were fundamental limits to our abilities to predict and control systems such as the weather
- Chaos Theory relates to the behaviour of systems across a wide range of scientific and non-scientific fields of study. As a result, progress depends largely on the rate at which experts in each of those fields integrate the ideas behind Chaos Theory into their own thinking.
Even in its media heyday, discussion of the Mandelbrot Set was restricted to its aesthetic qualities, to its ability to demonstrate the concept of self-similarity, and to the fact that it can be generated from very straightforward mathematical equations. Little attention was given to probably the most important point: that the Mandelbrot Set diagram describes a complex system which is sometimes stable (black areas) and sometimes unstable (blue and other colour areas), and where the boundary between these areas is highly complex. This at least suggests that the conditions which cause other systems, such as the weather or the economy, to move between stable and unstable states may also be extremely complex and difficult to master.
Watch a video demonstration of the beauty and self-similarity of the Mandelbrot Set below. Watch the second video to discover how the Mandelbrot Set is generated. This explains why the black areas are stable and the blue (and other colour) areas are unstable.
